Hi tbaleno,
I am growing Pumpkins in a drip system and in coarse sand/rice hull mix. My System is a prototype that combines drip and recycle and NFT but I believe there is not much difference with ordinary drip and recycle. My timer is adjusted to 15 minutes dripping every hour for night and morning. For late morning and afternoon it drips for 15 min. every 45 min.

One hour one day is not sufficient anyway, especially when your pumpkin plants grow bigger and will consume 1-2 gallons plus a day. In hydroton they will probably dry-out very quickly, depending on climate and temperatures (exposure) of course.

I have no experience with carrots in sand, but if the sand is deep enough and not too croase, it stocks much more humidity then hydroton. One hour a day may be sufficient, but here also, shorter and more dripping cycles would be better and highly recommended.

Yes, get a proper timer anyway, it's worth every dime ;-)